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                        <h1 class="single-title flipInX">热力学第一定律 等值过程 绝热过程</h1><div class="post-meta summary-post-meta"><span class="post-category meta-item">
                                <a href="/categories/%E7%83%AD%E5%8A%9B%E5%AD%A6/"><span class="svg-icon icon-folder"></span>热力学</a>
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                                <span class="svg-icon icon-clock"></span><time class="timeago" datetime="2020-05-30">2020-05-30</time>
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  <ul>
    <li><a href="#热力学第一定律">热力学第一定律</a>
      <ul>
        <li><a href="#内能功和热量">内能，功和热量</a></li>
        <li><a href="#准静态过程">准静态过程</a></li>
        <li><a href="#准静态过程的功与热">准静态过程的功与热</a>
          <ul>
            <li><a href="#体积功">体积功：</a></li>
            <li><a href="#准静态过程中的热量计算">准静态过程中的热量计算</a></li>
          </ul>
        </li>
        <li><a href="#热力学第一定律-1">热力学第一定律</a></li>
      </ul>
    </li>
    <li><a href="#理想气体等值过程">理想气体等值过程</a>
      <ul>
        <li><a href="#等容过程定容摩尔热容">等容过程，定容摩尔热容</a></li>
        <li><a href="#等压过程定压摩尔热容">等压过程，定压摩尔热容</a></li>
        <li><a href="#等温过程">等温过程</a></li>
        <li><a href="#绝热过程">绝热过程</a>
          <ul>
            <li><a href="#绝热方程">绝热方程</a></li>
            <li><a href="#绝热线与等温线">绝热线与等温线</a></li>
            <li><a href="#绝热过程中功值计算">绝热过程中功值计算</a></li>
          </ul>
        </li>
      </ul>
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                    </div><h2 id="热力学第一定律" class="headerLink"><a href="#%e7%83%ad%e5%8a%9b%e5%ad%a6%e7%ac%ac%e4%b8%80%e5%ae%9a%e5%be%8b" class="header-mark"></a>热力学第一定律</h2><h3 id="内能功和热量" class="headerLink"><a href="#%e5%86%85%e8%83%bd%e5%8a%9f%e5%92%8c%e7%83%ad%e9%87%8f" class="header-mark"></a>内能，功和热量</h3><ul>
<li>
<p>实际气体内能：所有热分子热运动的动能和分子势能的总和</p>
</li>
<li>
<p>内能是状态量: $E=E(T,V)$</p>
<p>理想气体内能: $E={\frac{M}{M_{mol}}{\frac{i}{2}}RT}$</p>
<p>是状态参量T的单值函数</p>
</li>
<li>
<p>系统内能改变的两种方式</p>
</li>
</ul>
<ol>
<li>做工可以改变系统的状态：摩擦升温（机械功），电加热（电功）</li>
<li>热量的传递可以改变系统的内能：热量是过程量</li>
</ol>
<h3 id="准静态过程" class="headerLink"><a href="#%e5%87%86%e9%9d%99%e6%80%81%e8%bf%87%e7%a8%8b" class="header-mark"></a>准静态过程</h3><div>
$$
热力学过程 = \left\{
  \begin{array}{lr}
    准静态过程\\
    非静态过程
  \end{array}
\right.
$$
</div>
<ul>
<li>准静态过程：系统从一个平衡态到另一个平衡态，如果过程中所有的中间态都可以近似的看作平衡态法过程</li>
<li>准静态过程是理想化过程</li>
</ul>
<p>




<figure class="render-image"><a target="_blank" href="https://img-blog.csdnimg.cn/20200529221714218.png" title=" " >
        <img loading="lazy" decoding="async"
             class="render-image"
             src="https://img-blog.csdnimg.cn/20200529221714218.png"
            alt="平衡态"
        />
    </a><figcaption class="image-caption"> </figcaption>
</figure></p>
<p><font color=red>弛豫时间$\tau$: </font>系统从一个平衡态变道相邻平衡态所经过的时间</p>
<p>当<font color=red>$\Delta t_{过程}&raquo;\tau$: </font>过程就可以视为准静态过程，故 **无限缓慢** 只是一个相对的概念。</p>
<p><font color=red>非静态过程: </font>系统从一平衡态到另一平衡态，过程中所有中间态为非静态的过程</p>
<ul>
<li>准静态过程曲线</li>
</ul>
<p>




<figure class="render-image"><a target="_blank" href="https://img-blog.csdnimg.cn/20200529222936128.png" title=" " >
        <img loading="lazy" decoding="async"
             class="render-image"
             src="https://img-blog.csdnimg.cn/20200529222936128.png"
            alt="准静态过程曲线"
        />
    </a><figcaption class="image-caption"> </figcaption>
</figure></p>
<p>p-V图上，一个点代表一个平衡态，一条连续的曲线代表一个准静态过程</p>
<h3 id="准静态过程的功与热" class="headerLink"><a href="#%e5%87%86%e9%9d%99%e6%80%81%e8%bf%87%e7%a8%8b%e7%9a%84%e5%8a%9f%e4%b8%8e%e7%83%ad" class="header-mark"></a>准静态过程的功与热</h3><p>




<figure class="render-image"><a target="_blank" href="https://img-blog.csdnimg.cn/20200529223844932.png" title=" " >
        <img loading="lazy" decoding="async"
             class="render-image"
             src="https://img-blog.csdnimg.cn/20200529223844932.png"
            alt=" "
        />
    </a><figcaption class="image-caption"> </figcaption>
</figure></p>
<h4 id="体积功" class="headerLink"><a href="#%e4%bd%93%e7%a7%af%e5%8a%9f" class="header-mark"></a>体积功：</h4><p>当活塞移动微小位移$dl$时，系统外界所做的元功为：</p>
<p>$$dA = Fdl = pSdl = pdV$$</p>
<p>$$A=\begin{aligned}\int_{V_{1}}^{V_{2}} p  \mathrm{d} V\end{aligned}$$</p>
<p>$dV&gt;0,dA&gt;0$系统对外界做正功</p>
<p>$dV&lt;0,dA&lt;0$系统对外界做负功</p>
<p>$dV=0,dA=0$系统不做功</p>
<ul>
<li>功是过程量</li>
<li>做功改变系统热力学状态的微观实质</li>
</ul>
<p>




<figure class="render-image"><a target="_blank" href="https://img-blog.csdnimg.cn/20200529225016265.png" title=" " >
        <img loading="lazy" decoding="async"
             class="render-image"
             src="https://img-blog.csdnimg.cn/20200529225016265.png"
            alt=" "
        />
    </a><figcaption class="image-caption"> </figcaption>
</figure></p>
<p>-功是系统与外界交换的能量的量度</p>
<h4 id="准静态过程中的热量计算" class="headerLink"><a href="#%e5%87%86%e9%9d%99%e6%80%81%e8%bf%87%e7%a8%8b%e4%b8%ad%e7%9a%84%e7%83%ad%e9%87%8f%e8%ae%a1%e7%ae%97" class="header-mark"></a>准静态过程中的热量计算</h4><p>$$C = \frac{dQ}{dT}$$</p>
<p>C（热容量）：系统在某一无限小过程中吸收热量$dQ$与温度变化$dT$的比值
单位：$J\cdot K^{-1}$</p>
<p>热容量与比热的关系为：$C = Mc_{比}$</p>
<p>$C_m$（摩尔热容量）：</p>
<p>$$C = {\frac{M}{M_{mol}}}{C_{m}}$$</p>
<p>$$dQ = {\frac{M}{M_{mol}}}{C_m}{dT}$$</p>
<p>$$Q = {\frac{M}{M_{mol}}}{C_m}(T_2-T_1)$$</p>
<ul>
<li>传热的微观本质：</li>
</ul>
<p>




<figure class="render-image"><a target="_blank" href="https://img-blog.csdnimg.cn/20200529230247149.png" title=" " >
        <img loading="lazy" decoding="async"
             class="render-image"
             src="https://img-blog.csdnimg.cn/20200529230247149.png"
            alt=" "
        />
    </a><figcaption class="image-caption"> </figcaption>
</figure></p>
<ul>
<li>热量也是能量变化的量度</li>
</ul>
<h3 id="热力学第一定律-1" class="headerLink"><a href="#%e7%83%ad%e5%8a%9b%e5%ad%a6%e7%ac%ac%e4%b8%80%e5%ae%9a%e5%be%8b-1" class="header-mark"></a>热力学第一定律</h3><p>对于任一过程，系统与外界可能同时有功和热量的转换，且系统能量改变仅为内能时，根据能量守恒：
$$\Delta E = Q + (-A)$$</p>
<p>或$$Q = \Delta E + A$$</p>
<ul>
<li>
<p>$Q&gt;0$系统吸热，$Q&lt;0$系统放热</p>
</li>
<li>
<p>$A&gt;0$系统对外做功，$A&lt;0$外界对系统做功</p>
</li>
<li>
<p>$\Delta E&gt; 0$系统内能增加，$\Delta E&lt;0$系统内能减少</p>
</li>
<li>
<p>如果系统经历一些微小变化过程，则$dQ=dE+dA$；</p>
</li>
<li>
<p>对准静态过程：</p>
</li>
</ul>
<p>$$dQ=dE+pdV$$</p>
<p>$$Q=\Delta E + {\begin{aligned}{\int_{V_{1}}^{V_{2}}}p{\mathrm{d} V}\end{aligned}}$$</p>
<h2 id="理想气体等值过程" class="headerLink"><a href="#%e7%90%86%e6%83%b3%e6%b0%94%e4%bd%93%e7%ad%89%e5%80%bc%e8%bf%87%e7%a8%8b" class="header-mark"></a>理想气体等值过程</h2><h3 id="等容过程定容摩尔热容" class="headerLink"><a href="#%e7%ad%89%e5%ae%b9%e8%bf%87%e7%a8%8b%e5%ae%9a%e5%ae%b9%e6%91%a9%e5%b0%94%e7%83%ad%e5%ae%b9" class="header-mark"></a>等容过程，定容摩尔热容</h3><p>$$\because dV=0,dA= pdV = 0$$
$$\therefore dQ=dE={\frac{M}{M_{mol}}}{\frac{i}{2}}RdT$$
$$Q_V=E_2-E_1={\frac{M}{M_{mol}}}{\frac{i}{2}}Rd(T_2-T_1)$$





<figure class="render-image"><a target="_blank" href="https://img-blog.csdnimg.cn/20200529232847856.png" title=" " >
        <img loading="lazy" decoding="async"
             class="render-image"
             src="https://img-blog.csdnimg.cn/20200529232847856.png"
            alt=" "
        />
    </a><figcaption class="image-caption"> </figcaption>
</figure></p>
<p><strong><font color=blue>定容摩尔热容量</font></strong></p>
<div>
$$dQ_V=dE={\frac{i}{2}}RdT$$
$$C_V=({\frac {dQ}{dT}})_V$$
$$C_{V,m}={\frac{i}{2}}R$$
</div>
<ul>
<li>
<p>单原子理想气体：$C_{V,m}={\frac{3}{2}}R$</p>
</li>
<li>
<p>双原子理想气体：$C_{V,m}={\frac{5}{2}}R$</p>
</li>
<li>
<p>多原子理想气体：$C_{V,m}=3R$</p>
</li>
</ul>
<p><strong><font color=blue>理想气体内能</font></strong>
$$E={\frac{M}{M_{mol}}}{C_{V,m}}T$$
理想气体的任一$T_1\rightarrow T_2$过程
<font color=red>$$dE=\nu C_{V,m}dT$$</font>
$$\Delta E=E_2-E_1={\nu}{\begin{aligned}{\int_{T_{1}}^{T_{2}}}{C_{V,m}}{\mathrm{d} T}\end{aligned}}$$
若$C_{V,m}$近似为常数，则有<font color=red>$\Delta E = \nu C_{V,m}\Delta T$</font></p>
<h3 id="等压过程定压摩尔热容" class="headerLink"><a href="#%e7%ad%89%e5%8e%8b%e8%bf%87%e7%a8%8b%e5%ae%9a%e5%8e%8b%e6%91%a9%e5%b0%94%e7%83%ad%e5%ae%b9" class="header-mark"></a>等压过程，定压摩尔热容</h3><p>$$dA=pdV$$
$$dQ_p=dE+pdV$$
$$A_p={\begin{aligned}{\int_{V_{1}}^{V_{2}}}p{\mathrm{d} V}\end{aligned}}=p(V_2-V_1)$$
<font color=red>$$Q_p={\frac{M}{M_{mol}}}{\frac{i}{2}}R(T_2-T_1)+{\frac{M}{M_{mol}}}R(T_2-T_1)$$</font></p>
<p>




<figure class="render-image"><a target="_blank" href="https://img-blog.csdnimg.cn/20200530001347196.png" title=" " >
        <img loading="lazy" decoding="async"
             class="render-image"
             src="https://img-blog.csdnimg.cn/20200530001347196.png"
            alt=" "
        />
    </a><figcaption class="image-caption"> </figcaption>
</figure></p>
<p><strong><font color=blue>定压摩尔热容量</font></strong>
$$dQ_p=dE+dA_p=C_{V,m}dT+pdV$$
$$pV=RT微分得pdV=RdT$$
$$dQ_p={\frac{i}{2}}R\cdot dT+R\cdot dT$$
<font color=red>$$C_{p,m}=(\frac{dQ}{dT})_p$$</font>
$$C_{p,m}={\frac{i}{2}}R+R$$
$$C_{p,m}=C_{V,m}+R$$
<font color=red>$$Q_{p,m}={\frac{M}{M_{mol}}}{C_{p,m}}(T_2-T_1)$$</font></p>
<p><strong>比热容比：</strong> $\gamma =\frac{C_{p,m}}{C_{V,m}}$为绝热系数</p>
<p>理想气体：$\gamma =\frac{C_{p,m}}{C_{V,m}}=\frac{\frac{i}{2}R+R}{\frac{i}{2}R}=\frac{i+2}{i}$</p>
<ul>
<li>对单原子分子：$i=3,\gamma=1.67$</li>
<li>对刚性双原子分子：$i=5,\gamma=1.40$</li>
<li>对刚性多原子分子：$i=6,\gamma=1.33$</li>
</ul>
<h3 id="等温过程" class="headerLink"><a href="#%e7%ad%89%e6%b8%a9%e8%bf%87%e7%a8%8b" class="header-mark"></a>等温过程</h3><p>$dT=0,dE=0$</p>
<p>$dQ_T=dA_T$</p>
<p>$dQ_T=pdV,p=\nu RT\cdot \frac{1}{V}$</p>
<p>$Q_T=A_T={\begin{aligned}{\int_{V_{1}}^{V_{2}}}\nu RT{\frac{dV}{V}}\end{aligned}}=\nu RTln{\frac{V_2}{V_1}}=p_1 V_1 ln{\frac{V_2}{V_1}}$</p>
<div>
$\Rightarrow Q_T = \left\{\begin{array}{lr}p_1 V_1 ln{\frac{p_1}{p_2}}=p_2 V_2 ln{\frac{p_1}{p_2}}\\\frac{M}{M_{mol}}RTln{\frac{p_1}{p_2}}\end{array}\right.$
</div>
<p>




<figure class="render-image"><a target="_blank" href="https://img-blog.csdnimg.cn/20200530081653297.png" title=" " >
        <img loading="lazy" decoding="async"
             class="render-image"
             src="https://img-blog.csdnimg.cn/20200530081653297.png"
            alt=" "
        />
    </a><figcaption class="image-caption"> </figcaption>
</figure></p>
<h3 id="绝热过程" class="headerLink"><a href="#%e7%bb%9d%e7%83%ad%e8%bf%87%e7%a8%8b" class="header-mark"></a>绝热过程</h3><p>系统变化过程中，系统与外界没有热交换</p>
<ul>
<li>特征：$dQ=0,dE+dA=0$</li>
</ul>
<h4 id="绝热方程" class="headerLink"><a href="#%e7%bb%9d%e7%83%ad%e6%96%b9%e7%a8%8b" class="header-mark"></a>绝热方程</h4><ul>
<li>
<p>对于准静态方程
$\nu C_{V,m}dT+pdV=0$</p>
<p>$pV=\nu RT$</p>
<p>取微分得</p>
<p>$pdV+Vdp=\nu RdT$</p>
<p>消去$\nu dT$得</p>
<p>$pdV+Vdp=-R{\frac{pdV}{C_{V,m}}}$</p>
<p>${C_{V,m}}pdV+{C_{V,m}}Vdp=-RpdV$</p>
<p>${C_{p,m}}pdV+{C_{V,m}}Vdp=0$</p>
<p>${\frac{dp}{p}}+\gamma {\frac{dV}{V}}=0$</p>
<p>积分得</p>
<p>${\begin{aligned}\int \frac{dp}{p}\end{aligned}}+{\begin{aligned}\int  \gamma \frac{dV}{V}\end{aligned}}=0$</p>
<p>得</p>
<p>$lnp+\gamma lnV=C$</p>
<p><font color=red>$lnpV^\gamma=C$</font></p>
<p><font color=red>$pV^\gamma=C_1$</font></p>
<p><font color=red>$pV^{\gamma-1}=C_2$</font></p>
<p><font color=red>$p^{\gamma-1}T^{-\gamma}=C_3$</font>,即松柏方程</p>
</li>
</ul>
<h4 id="绝热线与等温线" class="headerLink"><a href="#%e7%bb%9d%e7%83%ad%e7%ba%bf%e4%b8%8e%e7%ad%89%e6%b8%a9%e7%ba%bf" class="header-mark"></a>绝热线与等温线</h4><p>




<figure class="render-image"><a target="_blank" href="https://img-blog.csdnimg.cn/20200530101437801.png" title=" " >
        <img loading="lazy" decoding="async"
             class="render-image"
             src="https://img-blog.csdnimg.cn/20200530101437801.png"
            alt=" "
        />
    </a><figcaption class="image-caption"> </figcaption>
</figure></p>
<p>$pV=C_1,等温线$</p>
<p>$pV^\gamma=C_2,绝热线$</p>
<ul>
<li>
<p>对于等温过程</p>
<p>$pV=C_1=p_A V_A$</p>
<p>$p=\frac{C_1}{V}$</p>
<p>$\frac{dp}{dV}|_{AT}=-\frac{C_1}{V^2}|_A=-\frac{C_1}{V_A  ^2}=-\frac{p_AV_A}{V_A ^2}=-\frac{p_A}{V_A}$</p>
</li>
<li>
<p>对于绝热过程</p>
<p>$pV^\gamma=C_2=p_AV_A ^\gamma$</p>
<p>$p=\frac{C_2}{V^\gamma}$</p>
<p>$\frac{dp}{dV}|_{A\gamma}=-\gamma \frac{C_2}{V^{\gamma+1}}|_A=-\gamma \frac{p_AV_A ^\gamma}{V_A ^{\gamma+1}}=-\gamma \frac{p_A}{V_A}$</p>
<p>$\because \gamma &gt; 1$</p>
<p>$\therefore |\frac{dp}{dV}|<em>{A\gamma}=\gamma \frac{p_A}{V_A}&gt;|\frac{dp}{dV}|</em>{AT}=\frac{p_A}{V_A}$</p>
<p>即绝热线要陡一些</p>
</li>
</ul>
<p>$p=nkT$</p>
<p>




<figure class="render-image"><a target="_blank" href="https://img-blog.csdnimg.cn/2020053010365533.png" title=" " >
        <img loading="lazy" decoding="async"
             class="render-image"
             src="https://img-blog.csdnimg.cn/2020053010365533.png"
            alt=" "
        />
    </a><figcaption class="image-caption"> </figcaption>
</figure></p>
<p>




<figure class="render-image"><a target="_blank" href="https://img-blog.csdnimg.cn/20200530104130742.png" title=" " >
        <img loading="lazy" decoding="async"
             class="render-image"
             src="https://img-blog.csdnimg.cn/20200530104130742.png"
            alt=" "
        />
    </a><figcaption class="image-caption"> </figcaption>
</figure></p>
<h4 id="绝热过程中功值计算" class="headerLink"><a href="#%e7%bb%9d%e7%83%ad%e8%bf%87%e7%a8%8b%e4%b8%ad%e5%8a%9f%e5%80%bc%e8%ae%a1%e7%ae%97" class="header-mark"></a>绝热过程中功值计算</h4><p>




<figure class="render-image"><a target="_blank" href="https://img-blog.csdnimg.cn/20200530104351521.png" title=" " >
        <img loading="lazy" decoding="async"
             class="render-image"
             src="https://img-blog.csdnimg.cn/20200530104351521.png"
            alt=" "
        />
    </a><figcaption class="image-caption"> </figcaption>
</figure></p>
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